Communication Over MIMO Broadcast Channels Using Lattice-BasisReduction1

Mahmoud Taherzadeh, Amin Mobasher, and Amir K. Khandani

Coding & Signal Transmission Laboratory

Department of Electrical & Computer Engineering

University of Waterloo

Waterloo, Ontario, Canada, N2L 3G1Abstract

A new viewpoint for adopting the lattice reduction in communication over MIMO broadcast

channels is introduced. Lattice basis reduction helps us to reduce the average transmitted energy

by modifying the region which includes the constellation points. The new viewpoint helps us to

generalize the idea of lattice-reduction-aided precoding for the case of unequal-rate transmission,

and obtain analytic results for the asymptotic behavior (SNR−→∞) of the symbol-error-rate for the

lattice-reduction-aided precoding and the perturbation technique. Also, the outage probability for

both cases of fixed-rate users and fixed sum-rate is analyzed. It is shown that the lattice-reduction-

aided method, using LLL algorithm, achieves the optimum asymptotic slope of symbol-error-rate

(called the precoding diversity).

I. INTRODUCTION

In the recent years, communications over multiple-antenna fading channels has attracted

the attention of many researchers. Initially, the main interest has been on the point-to-point

Multiple-Input Multiple-Output (MIMO) communications [1]–[5]. In [1] and [2], the authors

have shown that the capacity of a MIMO point-to-point channel increases linearly with the

minimum number of the transmit and the receive antennas.

More recently, new information theoretic results [6], [7], [8], [9] have shown that

in multiuser MIMO systems, one can exploit most of the advantages of multiple-antenna

systems. It has been shown that in a MIMO broadcast system, the sum-capacity grows1This work was supported in part by funding from Communications and Information Technology Ontario (CITO), Nortel

Networks, and Natural Sciences and Engineering Research Council of Canada (NSERC). The material of this paper was

presented at the IEEE International Symposium on Information Theory, Adelaide, Australia, September 2005.

2

linearly with the minimum number of the transmit and receive antennas [7], [8], [9]. To

achieve the sum capacity, some information theoretic schemes, based on dirty-paper coding,

are introduced. Dirty-paper coding was originally proposed for the Gaussian interference

channel when the interfering signal is known at the transmitter [10]. Some methods, such

as using nested lattices, are introduced as practical techniques to achieve the sum-capacity

promised by the dirty-paper coding [11]. However, these methods are not easy to implement.

As a simple precoding scheme for MIMO broadcast systems, the channel inversion

technique (or zero-forcing beamforming [6]) can be used at the transmitter to separate the data

for different users. To improve the performance of the channel inversion technique, a zero-

forcing approximation of the dirty paper coding (based on QR decomposition) is introduced

in [6] (which can be seen as a scalar approximation of [11]). However, both of these methods

are vulnerable to the poor channel conditions, due to the occasional near-singularity of the

channel matrix (when the channel matrix has at least one small eigenvalue). This drawback

results in a poor performance in terms of the symbol-error-rate for the mentioned methods

[12].

In [12], the authors have introduced a vector perturbation technique which has a good

performance in terms of symbol error rate. Nonetheless, this technique requires a lattice

decoder which is an NP-hard problem. To reduce the complexity of the lattice decoder, in

[13]–[16], the authors have used lattice-basis reduction to approximate the closest lattice

point (using Babai approximation).

In this paper, we present a new viewpoint for the MIMO broadcast channel based on the

lattice-basis reduction. Instead of approximating the closest lattice point in the perturbation

problem, we use the lattice-basis reduction to reduce the average transmitted energy by

reducing the second moment of the fundamental region generated by the lattice basis. As we

will see later, this viewpoint helps us to: (i) achieve a better performance as compared to [14],

for the case that the data consists of odd integers (e.g. regular QAM constellations), (ii) extend

the idea for the case of unequal-rate transmission, and (iii) obtain some analytic results for the

asymptotic behavior (SNR−→∞) of the symbol-error-rate for both the proposed technique

and the perturbation technique of [12].

3

The rest of the paper is organized as the following: Sections II and III briefly describe

the system model and introduce the concept of lattice basis reduction. In section IV, the

proposed method is described and in section V, the proposed approach is extended for the

case of unequal-rate transmission. In section VI, we consider the asymptotic performance of

the proposed method for high SNR values, in terms of the probability of error. We define the

precoding diversity and the outage probability for the case of fixed-rate users. It is shown that

by using lattice basis reduction, we can achieve the maximum precoding diversity. For the

proof, we use a bound on the orthogonal deficiency of an LLL-reduced basis. Also, an upper

bound is given for the probability that the length of the shortest vector of a lattice (generated

by complex Gaussian vectors) is smaller than a given value. Using this result, we also show

that the perturbation technique achieves the maximum precoding diversity. In section VII,

some simulation results are presented. These results show that the proposed method offers

almost the same performance as [12] with a much smaller complexity. As compared to [14],

the proposed method offers almost the same performance. However, by sending a very small

amount of side information (a few bits for one fading block), the modified proposed method

offers a better performance with a similar complexity (detailed discussion about the relevance

with [14] is presented in section IV). Finally, in section VIII, some concluding remarks are

presented.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a multiple-antenna broadcast system with N transmit antennas and M

single-antenna users (N ≥M ). Consider y = [y1, ..., yM ]T , x = [x1, ..., xN ]T , w = [w1, ..., wM ]T ,

and the M × N matrix H, respectively, as the received signal, the transmitted signal, the

noise vector, and the channel matrix. The transmission over the channel can be formulated

as,

y = Hx + w. (1)

The channel is assumed to be Raleigh, i.e. the elements of H are i.i.d. with the zero-

mean unit-variance complex Gaussian distribution and the noise is i.i.d. additive Gaussian.

Moreover, we have the energy constraint on the transmitted signal, E(‖x‖2) = 1. The energy

4

of the additive noise is σ2 per antenna, i.e. E(‖w‖2) = Mσ2. The Signal-to-Noise Ratio

(SNR) is defined as ρ = 1σ2 .

In a broadcast system, the receivers do not cooperate with each other (they should

decode their respective data, independently). The main strategy in dealing with this restriction

is to apply an appropriate precoding scheme at the transmitter. The simplest method in this

category is using the channel inversion technique at the transmitter to separate the data for

different users:

s = H+u, (2)

where H+ = HH(HHH)−1, and HH is the Hermitian of H. Moreover, s is the transmitted

signal before the normalization (x =s

√

E(‖s‖2)is the normalized transmitted signal), and

u is the data vector, i.e. ui is the data for the i’th user. For N = M (the number of transmit

antennas and the number of users are equal), the transmitted signal is

s = H−1u. (3)

The problem arises when H is poorly conditioned and ‖s‖ becomes very large, resulting

in a high power consumption. This situation occurs when at least one of the singular values of

H is very small which results in vectors with large norms as the columns of H+. Fortunately,

most of the time (especially for high SNRs), we can combat the effect of a small singular

value by changing the supporting region of the constellation which is the main motivation

behind the current article.

When the data of different users are selected from Z[i], the overall constellation can

be seen as a set of lattice points. In this case, lattice algorithms can be used to modify

the constellation. Especially, lattice-basis reduction is a natural solution for modifying the

supporting region of the constellation.

III. LATTICE-BASIS REDUCTION

Lattice structures have been frequently used in different communication applications

such as quantization or decoding of MIMO systems. A real (or complex) lattice Λ is a

discrete set of N -D vectors in the real Euclidean space RN (or the complex Euclidean space

5

CN ) that forms a group under ordinary vector addition. Every lattice Λ is generated by the

integer linear combinations of some set of linearly independent vectors b1, · · · ,bM ∈ Λ,

where the integer M , M ≤ N , is called the dimension of the lattice Λ. The set of vectors

{b1, · · · ,bM} is called a basis of Λ, and the matrix B = [b1, · · · ,bM ], which has the basis

vectors as its columns, is called the basis matrix (or generator matrix) of Λ.The basis for representing a lattice is not unique. Usually a basis consisting of relatively

short and nearly orthogonal vectors is desirable. The procedure of finding such a basis for

a lattice is called Lattice Basis Reduction. A popular criterion for lattice-basis reduction is

to find a basis such that ‖b1‖ · ... · ‖bM‖ is minimized. Because the volume of the lattice2

does not change with the change of basis, this problem is equivalent to minimizing the

orthogonality defect which is defined as

δ ,(‖b1‖2‖b2‖2...‖bM‖2)

det BHB. (4)

The problem of finding such a basis is NP-hard [17]. Several distinct sub-optimal

reductions have been studied in the literature, including those associated to the names

Minkowski, Korkin-Zolotarev, and more recently Lenstra-Lenstra and Lovasz (LLL) [18].An ordered basis (b1, · · · ,bM) is a Minkowski-Reduced Basis [19] if

• b1 is the shortest nonzero vector in the lattice Λ, and

• For each k = 2, ...,M , bk is the shortest nonzero vector in Λ such that (b1, · · · ,bk)may be extended to a basis of Λ.

Minkowski reduction can be seen as a greedy solution for the lattice-basis reduction problem.

However, finding Minkowski reduced basis is equivalent to finding the shortest vector in the

lattice and this problem by itself is NP-hard. Thus, there is no polynomial time algorithm

for this reduction method.In [20], a reduction algorithm (called LLL algorithm) is introduced which uses the

Gram-Schmidt orthogonalization and has a polynomial complexity and guarantees a bounded

orthogonality defect. For any ordered basis of Λ, say (b1, · · · ,bM), one can compute an

ordered set of Gram-Schmidt vectors,(

b̂1, · · · , b̂M)

, which are mutually orthogonal, using

the following recursion:2Volume of the lattice generated by B is

`

detBHB

´ 1

2 .

6

b̂i = bi −∑i−1

j=1 µijb̂j, with

µij =< bi, b̂j >

< bj, b̂j >.

(5)

where < ·, · > is the inner product. An ordered basis (b1, · · · ,bM) is an LLL Reduced Basis[20] if,

• ‖µij‖ ≤ 12

for 1 ≤ i < j ≤M , and

• p · ‖b̂i‖2 ≤ ‖b̂i+1 + µi+1,ib̂i‖2

where 14< p < 1, and

(

b̂1, · · · , b̂M)

is the Gram-Schmidt orthogonalization of the ordered

basis and bi =∑i

j=1 µijb̂j for i = 1, ...,M .

It is shown that LLL basis-reduction algorithm produces relatively short basis vectors

with a polynomial-time computational complexity [20]. The LLL basis reduction has found

extended applications in several contexts due to its polynomial-time complexity. In [21],

LLL algorithm is generalized for Euclidean rings (including the ring of complex integers).

In this paper, we will use the following important property of the complex LLL reduction

(for p = 34):

Theorem 1 (see [21]): Let Λ be an M -dimensional complex lattice and B = [b1...bM ]

be the LLL reduced basis of Λ. If δ is the orthogonality defect of B, then,√δ ≤ 2M(M−1). (6)

In the rest of this paper, all matrix computations are in complex domain and also for

the lattice-basis reduction, the complex LLL algorithm is used.

IV. PROPOSED APPROACH

Assume that the data for different users, ui, is selected from the points of the integer

lattice (or from the half-integer grid [22]). The data vector u is a point in the Cartesian

product of these sub-constellations. As a result, the overall receive constellation consists

of the points from Z2M , bounded within a 2M -dimensional hypercube. At the transmitter

side, when we use the channel inversion technique, the transmitted signal is a point inside

a parallelotope whose edges are parallel to vectors, defined by the columns of H+. If the

data is a point from the integer lattice Z2M , the transmitted signal is a point in the lattice

7

generated by H+. When the squared norm of at least one of the columns of H+ is too large,

some of the constellation points require high energy for the transmission. We try to reduce

the average transmitted energy, by replacing these points with some other points with smaller

square norms. However, the lack of cooperation among the users imposes the restriction that

the received signals should belong to the integer lattice Z2M (to avoid the interference among

the users). The core of the idea in this paper is based on using an appropriate supporting

region for the transmitted signal set to minimize the average energy, without changing the

underlying lattice. This is achieved through the lattice-basis reduction.

When we use the continuous approximation (which is appropriate for large constella-

tions), the average energy of the transmitted signal is approximated by the second moment

of the transmitted region [22]. When we assume equal rates for the users, e.g. R bits per

user (R2

bits per dimension), the signal points (at the receiver) are inside a hypercube with

an edge of length a where

a = 2R/2. (7)

Therefore, the supporting region of the transmitted signal is the scaled version of the fun-

damental region of the lattice generated by H+ (corresponding to its basis) with the scaling

factor a. Note that by changing the basis for this lattice, we can change the corresponding

fundamental region (a parallelotope generated by the basis of the lattice and centered at

the origin). The second moment of the resulting region is proportional to the sum of the

squared norms of the basis vectors (see Appendix A). Therefore, we should try to find a

basis reduction method which minimizes the sum of the squared norms of the basis vectors.

Figure 1 shows the application of the lattice basis reduction in reducing the average energy

by replacing the old basis with a new basis which has shorter vectors. In this figure, by

changing the basis a1, a2 (columns of H+) to b1,b2 (the reduced basis), the fundamental

region F , generated by the original basis, is replaced by F ′, generated by the reduced basis.

Among the known reduction algorithms, the Minkowski reduction can be considered

as an appropriate greedy algorithm for our problem. Indeed, the Minkowski algorithm is the

successively optimum solution because in each step, it finds the shortest vector. However,

the complexity of the Minkowski reduction is equal to the complexity of the shortest-lattice-

8

b2

b1

F

F′

a2

a1

Fig. 1. Using lattice-basis reduction for reducing the average energy

vector problem which is known to be NP-hard [23]. Therefore, we use the LLL reduction

algorithm which is a suboptimum solution with a polynomial complexity.Assume that B = H+U is the LLL-reduced basis for the lattice obtained by H+,

where U is an M ×M unimodular matrix (both U and U−1 have integer entries). We use

x = Bu′ = H+Uu′ as the transmitted signal where

u′ = U−1u mod a (8)

is the precoded data vector, u is the original data vector, and a is the length of the edges

of the hypercube, defined by (7). At the receiver side, we use modulo operation to find the

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original data:

y = Hx + n = HH+U(U−1u mod a) + n = U(U−1u mod a) + n (9)

= UU−1u mod a + n = u mod a + n. (10)

In obtaining (10) from (9), we use the fact that U and U−1 have integer entries.

In this method, at the beginning of each fading block, we reduce the lattice obtained

by H+ and during this block the transmitted signal is computed using (8). Neglecting the

preprocessing at the beginning of the block (for lattice reduction), the complexity of the

precoding is in the order of a matrix multiplication and a modulo operation. Therefore, the

complexity of the proposed precoding method is comparable to the complexity of the channel

inversion method. However, as we will show by the simulation results, the performance of

this method is significantly better, and indeed, is near the performance of the perturbation

method, presented in [12].

In the perturbation technique [12], the idea of changing the support region of the

constellation has been implemented using a different approach. In [12], u′ = u + al is

used as the precoded data, where the integer vector l is chosen to minimize ‖H+(u + al)‖.This problem is equivalent to the closest-lattice-point problem for the lattice generated by

aH+ (i.e. finding the lattice point which is closer to −H+u). Therefore, in the perturbation

technique, the support region of the constellation is a scaled version of the Voronoi region

[24] of the lattice. In the proposed method, we use a parallelotope (generated by the reduced

basis of the lattice), instead of the Voronoi region. Although this approximation results in

a larger second moment (i.e. higher energy consumption), it enables us to use a simple

precoding technique, instead of solving the closest-lattice-point problem.

For the lattice constellations, using a parallelotope instead of the Voronoi region (pre-

sented in this paper) is equivalent with using the Babai approximation instead of the exact

lattice decoding (previously introduced in [14]). In the case of using all lattice points inside

a region, the only practical difference between our lattice-reduction-aided scheme and the

scheme presented in [14] is that we reduce H+, while in [14], HH is reduced. This difference

10

has no significant effect on the performance. However, the new viewpoint helps us in

extending the proposed method for the case of variable-rate transmission and obtaining some

analytical results for the asymptotic performance.

The performance of the proposed lattice-reduction aided scheme can be improved by

combining it with other schemes, such as regularization [25] or the V-BLAST precoding

[26], or by sending a very small amount of side information. In the rest of this section, we

present two of these modifications.

A. Regularized lattice-reduction-aided precoding

In [25], the authors have proposed a regularization scheme to reduce the transmitted

power, by avoiding the near-singularity of H. In this method, instead of using H+ =

HH (HHH)−1, the transmitted vector is constructed as

x = HH(

HHH + αI)−1

u (11)

where α is a positive number. To combine the regularized scheme with our lattice-reduction-

aided scheme, we consider Br as the matrix corresponding to the reduced basis of the

lattice generated by HH (HHH + αI)−1. When we use the regularization, the received signals

of different users are not orthogonal anymore and the interference acts like extra noise.

Parameter α should be optimized such that the ratio between the power of received signal

and the power of the effective noise is minimized [25].

B. Modified lattice-reduction-aided precoding with small side information

In practical systems, we are interested in using a subset of points with odd coordinates

from the integer lattice. In these cases, we can improve the performance of the proposed

method by sending a very small amount of side information. When the data vector u

consists of odd integers, using the lattice-basis reduction may result in points with some even

coordinates (i.e. U−1u has some even elements), instead of points with all-odd coordinates

in the new basis. For this case, in (8), the set of precoded data u′ is not centered at the origin,

hence the transmitted constellation (which includes all the valid points Bu′) is not centered at

the origin. Therefore, we can reduce the transmitted energy and improve the performance by

11

shifting the center of the constellation to the origin. This situation is depicted in figure 2 for

a small constellation. For the sake of simplicity, this example is shown for a 2-dimensional

real space, but all the discussion in the paper are for complex vectors and matrices (In this

example, U−1 =

1 0

1 1

.). In comparison to the scheme presented in [14], which is the

lattice-reduction-aided approximation of the precoding scheme, our approach helps us to

utilize the fact that only a subset of lattice points (i.e. odd lattice points) are used in the

transmission.

It can be shown that the translation vector is equal to (U−1[1 + i, 1 + i, · · · , 1 + i]T +

[1 + i, 1 + i, · · · , 1 + i]T) mod 2 where i =√−1. When we use this shifted version of

the constellation, we must send the translation vector to the users (by sending 2 bits per

user) at the beginning of the block. However, compared to the size of the block of data, the

overhead of these two bits is negligible. Also, it should be noted that in all MIMO broadcast

schemes, at the beginning of the fading block we need to send the information about power

normalization.

The above idea of using a shift vector can be also used to improve the perturbation

technique (if we only use the odd points of the lattice). After reducing the inverse of the

channel matrix and obtaining the bits (corresponding to the shift vector) at the beginning of

each fading block, the closest point to the signal computed in equation (8) can be found by

using the sphere decoder. Then, the transmitted signal is obtained by

x = B (u′ + al + upar) , (12)

where upar is the zero-one shift vector, which is computed for users at the beginning of the

fading block, and the perturbation vector l is an even integer vector such that the vector x

has the minimum energy. This method which can be considered as modified perturbationmethod outperforms the perturbation method in [12]. When we are not restricted to the odd

lattice points, using (12) instead of H+ (u′ + al) does not change the performance of the

perturbation method. It only reduces the complexity of the lattice decoder [27].

12

Constellation for data u

Constellation for the precoded data u′

Points corresponding to U−1

u

Constellation for the shifted precoded data u′

shifted

Fig. 2. In the case of using odd lattice points, when U−1

u has some even entries, we can shift the precoded data to

reduce the average energy. For the sake of simplicity, this example is shown for a 2-dimensional real space, but all the

discussions in the paper are for complex vectors and matrices.

V. UNEQUAL-RATE TRANSMISSION

In the previous section, we had considered the case that the transmission rates for

different users are equal. In some applications, we are interested in assigning different rates

to different users. Consider R1, ..., RM as the transmission rates3 for the users (we consider

them as even integer numbers). Equation (8) should be modified as

u′ = U−1u mod a, (13)3R1, ..., RM are rates per complex dimension (rate per real dimension for different users are R1

2, ...,

RM

2)

13

where the entries of a = [a1, ..., aM ]T are equal to4

ai = 2Ri/2. (14)

Also, at the receiver side, instead of (9) and (10), we have

y = Hx + n = HH+U(U−1u mod a) + n = U(U−1u mod a) + n (15)

= UU−1u mod a + n = u mod a + n. (16)

If we are interested in sum-rate, instead of individual rates, we can improve the perfor-

mance of the proposed method by assigning variable rates to different users. We assume that

the sum-rate (rather than the individual rates) is fixed and we want to reduce the average

transmitted energy. To simplify the analysis, we use the continuous approximation which has

a good accuracy for high rates.

Considering continuous approximation, the sum-rate is proportional to the logarithm of

the volume of the lattice with basis B and the average energy is proportional to the second

moment of the corresponding parallelotope, which is proportional to∑M

i=1 ‖bi‖2 = trBBH

(see Appendix A). The goal is to minimize the average energy while the sum-rate is fixed.

We can use another lattice generated by B′ with the same volume, where its basis vectors

are scaled versions of the vectors of the basis B, according to different rates for different

users. Therefore, we can use B′ = BD instead of B (where D is a unit determinant M ×Mdiagonal matrix which does not change the volume of the lattice). For a given reduced basis

B, the product of the squared norms of the new basis vectors is constant:

‖b′1‖2‖b′2‖2...‖b′M‖2 = (‖b1‖2‖b2‖2...‖bM‖2) det D

= ‖b1‖2‖b2‖2...‖bM‖2 = const.(17)

The average energy corresponding to the new lattice basis should be minimized. When

we use the modified basis B′ instead of B, the average energy is proportional to∑M

i=1 ‖b′i‖2 =

trB′B′H (see Appendix A). According to the arithmetic-geometric mean inequality,∑M

i=1 ‖b′i‖2 =

trB′B′H is minimized iff

‖b′1‖ = ‖b′2‖ = ... = ‖b′M‖. (18)

4The data is assumed to be from Z[i]. With this assumption, to have rate Ri

2per real dimension, ai = 2

Ri

2 .

14

Therefore,

min trB′B′H = M(

‖b′1‖2‖b′2‖2...‖b′M‖2) 1M = M

(

‖b1‖2‖b2‖2...‖bM‖2) 1M (19)

Having the matrix B, the columns of matrix B′ can be found using the equation (18)

and trB′B′H can be obtained by (19). Now, for the selection of the reduced basis B, we should

find B such that ‖b1‖2‖b2‖2...‖bM‖2 is minimized. Because det BHB = det (H+)HH+ is

given, the best basis reduction is the reduction which maximizes ‖b1‖2‖b2‖2·...·‖bM‖2|det BHB| , or in

other words, minimizes the orthogonality defect.

In practice, we use discrete values for the rate, and sometimes, we should assign the rate

zero to some users (when their channel is very bad). In this case, for the rate assignment for

other users, we use the lattice reduction on the corresponding sublattice. It should be noted

that the average transmit power is fixed per channel realization and no long-run averaging

is considered, and no long-run power allocation is used. Also, the design criteria is guided

by continuous approximation, which is not appropriate for low rates and low SNR values.

However, as it is shown in the simulation results, the scheme works well.

VI. DIVERSITY AND OUTAGE PROBABILITY

In this section, we consider the asymptotic behavior (ρ −→ ∞) of the symbol error

rate (SER) for the proposed method and the perturbation technique. We show that for both

of these methods, the asymptotic slope of the SER curve is equal to the number of transmit

antennas. By considering the outage probability of a fixed-rate MIMO broadcast system,

we will show that for the SER curve in high SNR, the slope obtained by the proposed

method has the largest achievable value. Also, we analyze the asymptotic behavior of the

outage probability for the case of fixed sum-rate. We show that in this case, the slope of

the corresponding curve is equal to the product of the number of transmit antennas and the

number of single-antenna users.

A. Fixed-rate users

When we have the Channel-State Information (CSI) at the transmitter, without any

assumption on the transmission rates, the outage probability is not meaningful. However,

15

when we consider given rates R1, ..., RM for different users, we can define the outage

probability Pout as the probability that the point (R1, ..., RM) is outside the capacity region.

Theorem 2: For a MIMO broadcast system with N transmit antennas, M single-antenna

receivers (N ≥M ), and given rates R1, ..., RM ,

limρ→∞

− logPoutlog ρ

≤ N. (20)

Proof: Define Pout1 as the probability that the capacity of the point-to-point system

corresponding to the first user (consisting of N transmit antennas and one receive antenna

with independent channel coefficients and CSI at the transmitter) is less than R1:

Pout1 = Pr{log(

1 + ρ‖h1‖2)

≤ R1} (21)

where h1 is the vector defined by the first row of H. Note that the entries of h1 have iid

complex Gaussian distribution with unit variance. Thus, its square norm has a chi square

distribution. We have,

Pr{

log(

2ρ‖h1‖2)

≤ R1

}

(22)

= Pr

{

‖h1‖2 ≤ 2R1

2ρ

}

(23)

=

∫ 2R12ρ

0

f‖h1‖2(x) dx (24)

=

∫ 2R12ρ

0

1

(N − 1)!xN−1e−x dx (25)

We are interested in the large values of ρ. For ρ > 2R1−1,

∫ 2R12ρ

0

1

(N − 1)!xN−1e−x dx ≥

∫ 2R12ρ

0

1

(N − 1)!xN−1e−1 dx (26)

=e−1

(N − 1)!

∫ 2R12ρ

0

xN−1 dx (27)

=2NR1c

ρN(28)

16

where c = e−1

2NN !is a constant number. Now,

log(

1 + ρ‖h1‖2)

≤ log(

2ρ‖h1‖2)

for ρ >1

‖h1‖2(29)

=⇒ limρ→∞

− log Pr{log (1 + ρ‖h1‖2) ≤ R1}log ρ

(30)

≤ limρ→∞

− log Pr{log (2ρ‖h1‖2) ≤ R1}log ρ

(31)

≤ limρ→∞

− log 2NR1cρN

log ρ= N (32)

=⇒ limρ→∞

− logPout1log ρ

≤ N. (33)

According to the definition of Pout1, Pout ≥ Pout1. Therefore,

limρ→∞

− logPoutlog ρ

≤ limρ→∞

− logPout1log ρ

≤ N. (34)

We can define the diversity gain of a MIMO broadcast constellation or its precodingdiversity as limρ→∞

− logPelog ρ

where Pe is the probability of error. Similar to [28, lemma 5],

we can bound the precoding diversity by limρ→∞− logPout

log ρ. Thus, based on theorem 2, the

maximum achievable diversity is N .

We show that the proposed method (based on lattice-basis reduction) achieves the

maximum precoding diversity. To prove this, in lemma 1 and lemma 2, we relate the length

of the largest vector of the reduced basis B to dHH (the minimum distance of the lattice

generated by HH). In lemma 3, we bound the probability that dHH is too small. Finally,

in theorem 3, we prove the main result by relating the minimum distance of the receive

constellation to the length of the largest vector of the reduced basis B, and combining the

bounds on the probability that dHH is too small, and the probability that the noise vector is

too large.

17

Lemma 1: Consider B = [b1...bM ] as an N ×M matrix, with the orthogonality defect

δ, and B−H = [a1...aM ] as the inverse of its Hermitian (or its pseudo-inverse if M < N ).

Then,

max{‖b1‖, ..., ‖bM‖} ≤√δ

min{‖a1‖, ..., ‖aM‖}(35)

and

max{‖a1‖, ..., ‖aM‖} ≤√δ

min{‖b1‖, ..., ‖bM‖}. (36)

Proof: Consider bi as an arbitrary column of B. The vector bi can be written as

b′i+∑

i6=j ci,jbj , where b′i is orthogonal to bj for i 6= j. Now, [b1...bi−1b′ibi+1...bM ] can be

written as BP where P is a unit-determinant M ×M matrix (a column operation matrix):

‖b1‖2...‖bi−1‖2.‖bi‖2.‖bi+1‖2...‖bM‖2 (37)

= δ det BHB = δ det PHBHBP (38)

= δ det(

[b1...bi−1b′ibi+1...bM ]H[b1...bi−1b

′ibi+1...bM ]

)

. (39)

According to the Hadamard theorem:

det(

[b1...bi−1b′ibi+1...bM ]H[b1...bi−1b

′ibi+1...bM ]

)

≤ (40)

‖b1‖2...‖bi−1‖2.‖b′i‖2.‖bi+1‖2...‖bM‖2. (41)

Therefore,

‖b1‖2...‖bi−1‖2.‖bi‖2.‖bi+1‖2...‖bM‖2 ≤ δ‖b1‖2...‖bi−1‖2.‖b′i‖2.‖bi+1‖2...‖bM‖2 (42)

=⇒ ‖bi‖ ≤√δ‖b′i‖. (43)

Also, B+B = I results in <ai,bi> = 1 and <ai,bj> = 0 for i 6= j. Therefore,

1 = <ai,bi> = <ai, (b′i +∑

i6=jci,jbj)> = <ai,b

′i> (44)

Now, ai and b′i, both are orthogonal to the (M − 1)-dimensional subspace generated by the

vectors bj (j 6= i). Thus,

1 = <ai,b′i> = ‖ai‖.‖b′i‖ ≥ ‖ai‖.

‖bi‖√δ

(45)

18

=⇒ 1 ≥ ‖bi‖.‖ai‖√δ

(46)

=⇒ ‖bi‖ ≤√δ

‖ai‖(47)

The above relation is valid for every i, 1 ≤ i ≤ M . Without loss of generality, we can

assume that max{‖b1‖, ..., ‖bM‖} = ‖bk‖:

max{‖b1‖, ..., ‖bM‖} = ‖bk‖ ≤√δ

‖ak‖(48)

≤√δ

min{‖a1‖, ..., ‖aM‖}. (49)

Similarly, by using (47), we can also obtain the following inequality:

max{‖a1‖, ..., ‖aM‖} ≤√δ

min{‖b1‖, ..., ‖bM‖}. (50)

Lemma 2: Consider B = [b1...bM ] as an LLL-reduced basis for the lattice generated

by H+ and dHH as the minimum distance of the lattice generated by HH. Then, there is a

constant αM (independent of H) such that

max{‖b1‖, ..., ‖bM‖} ≤αMdHH

. (51)

Proof: According to the theorem 1,√δ ≤ 2M(M−1). (52)

Consider B−H = [a1, ..., aM ]. By using lemma 1 and (52),

max{‖b1‖, ..., ‖bM‖} ≤√δ

min{‖a1‖, ..., ‖aM‖}≤ 2M(M−1)

min{‖a1‖, ..., ‖aM‖}(53)

The basis B can be written as B = H+U for some unimodular matrix U:

B−H = ((H+U)H)+ = (UHH−H)+ = HHU−H. (54)

Noting that U−H is unimodular, B−H = [a1, ..., aM ] is another basis for the lattice generated

by HH. Therefore, the vectors a1, ..., aM are vectors from the lattice generated by HH, and

therefore, the length of each of them is at least dHH :

‖ai‖ ≥ dHH for 1 ≤ i ≤M (55)

19

=⇒ min{‖a1‖, ..., ‖aM‖} ≥ dHH (56)

(53) and (56) =⇒ max{‖b1‖, ..., ‖bM‖} ≤2M(M−1)

dHH

. (57)

Lemma 3: Assume that the entries of the N ×M matrix H has independent complex

Gaussian distribution with zero mean and unit variance and consider dH as the minimum

distance of the lattice generated by H. Then, there is a constant βN,M such that

Pr {dH ≤ ε} ≤

βN,Mε2N for M < N

βN,Nε2N .max

{

−(ln ε)N+1, 1}

for M = N. (58)

Proof: See Appendix B.

Theorem 3: For a MIMO broadcast system with N transmit antennas and M single-

antenna receivers (N ≥ M ) and fixed rates R1, ..., RM , using the lattice-basis-reduction

method,

limρ→∞

− logPelog ρ

= N. (59)

Proof: Consider B = [b1...bM ] as the LLL-reduced basis for the lattice generated by

H+. Each transmitted vector s is inside the parallelotope, generated by r1b1, ..., rMbM (where

r1, ..., rM are constant values determined by the rates of the users). Thus, every transmitted

vector s can be written as

s = t1b1 + ... + tMbM ,−ri2≤ ti ≤

ri2. (60)

For each of the transmitted vectors, the energy is

P = ‖s‖2 = ‖t1b1 + ... + tMbM‖2 (61)

=⇒ P ≤ (‖t1b1‖+ ...+ ‖tMbM‖)2 (62)

=⇒ P ≤(r1

2‖b1‖+ ... +

rM2‖bM‖

)2

. (63)

Thus, the average transmitted energy is

Pav = E(P ) ≤M2(

max{r1

2‖b1‖, ...,

rM2‖bM‖

})2

≤ c1.(max{‖b1‖2, ..., ‖bM‖2}) (64)

20

where c1 = M2

4max {r2

1, ..., r2M}. The received signals (without the effect of noise) are points

from the Z2M lattice. If we consider the normalized system (by scaling the signals such that

the average transmitted energy becomes equal to one),

d2 =1

Pav≥ 1

c1.(max{‖b1‖2, ..., ‖bM‖2}) (65)

is the squared distance between the received signal points.

For the normalized system, 1

ρis the energy of the noise at each receiver and 1

2ρis the

energy of the noise per each real dimension. Using (65), for any positive number γ,

Pr

{

d2 ≤ γ

ρ

}

≤ Pr

{

1

c1 max{‖b1‖2, ..., ‖bM‖2} ≤γ

ρ

}

(66)

Using lemma 2,

max{‖b1‖, ..., ‖bM‖} ≤αMdHH

(67)

(66), (67) =⇒ Pr

{

d2 ≤ γ

ρ

}

≤ Pr

{

d2HH

c1α2M

≤ γ

ρ

}

= Pr

{

d2HH ≤ γc1α

2M

ρ

}

(68)

The N ×M matrix HH has independent complex Gaussian distribution with zero mean

and unit variance. Therefore, by using lemma 3, we can bound the probability that dHH is

too small.

Case 1, M = N :

Pr

{

d2 ≤ γ

ρ

}

≤ Pr

{

d2HH ≤ γc1α

2N

ρ

}

(69)

≤ βN,N

(

γc1α2N

ρ

)N

max

{

(

−1

2lnγc1α

2N

ρ

)N+1

, 1

}

(70)

21

≤ βN,N

(

γc1α2N

ρ

)N

max{

(ln ρ)N+1 , 1}

for γ > 1 and ρ >1

c1α2N

(71)

≤ c2γN

ρN(ln ρ)N+1 for γ > 1 and ρ > max

{

1

c1α2N

, e

}

(72)

where c2 is a constant number and e is the Euler number.

If the magnitude of the noise component in each real dimension is less than 12d, the

transmitted data will be decoded correctly. Thus, we can bound the probability of error by

the probability that |wi|2 is greater than 14d2 for at least one i, 1 ≤ i ≤ 2N . Therefore, using

the union bound,

Pe ≤ 2N

(

Pr

{

|w1|2 ≥1

4d2

})

(73)

= 2N

(

Pr

{

d2 ≤ 4

ρ

}

.Pr

{

|w1|2 ≥1

4d2

∣

∣

∣

∣

d2 ≤ 4

ρ

}

+ Pr

{

4

ρ≤ d2 ≤ 8

ρ

}

.Pr

{

|w1|2 ≥1

4d2

∣

∣

∣

∣

4

ρ≤ d2 ≤ 8

ρ

}

+ Pr

{

8

ρ≤ d2 ≤ 16

ρ

}

.Pr

{

|w1|2 ≥1

4d2

∣

∣

∣

∣

8

ρ≤ d2 ≤ 16

ρ

}

+ ...

)

(74)

≤ 2N

(

Pr

{

d2 ≤ 4

ρ

}

+ Pr

{

4

ρ≤ d2 ≤ 8

ρ

}

.Pr

{

|w1|2 ≥1

4.4

ρ

}

+ Pr

{

8

ρ≤ d2 ≤ 16

ρ

}

.Pr

{

|w1|2 ≥1

4.8

ρ

}

+ ...

)

(75)

≤ 2N

(

Pr

{

d2 ≤ 4

ρ

}

+ Pr

{

d2 ≤ 8

ρ

}

.Pr

{

|w1|2 ≥1

ρ

}

+ Pr

{

d2 ≤ 16

ρ

}

.Pr

{

|w1|2 ≥2

ρ

}

+ ...

)

(76)

For the product terms in (76), we can bound the first part by (72). To bound the second

part, we note that w1 has real Gaussian distribution with variance 12ρ

. Therefore,

Pr

{

|w1|2 ≥θ

ρ

}

= Q(√

2θ) ≤ e−θ (77)

22

Now, for ρ > max{

1c1α2

N, e}

,

(72), (76) and (77) =⇒ Pe ≤ 2N

(

Pr

{

|w1|2 ≥1

4d2

})

(78)

≤ 2N

(

4Nc2

ρN(ln ρ)N+1 +

∞∑

i=0

2N(i+3)c2

ρN(ln ρ)N+1 e−2i

)

(79)

≤ (ln ρ)N+1

ρN.c2.2N

(

4N +

∞∑

i=0

2N(i+3)e−2i

)

(80)

≤ c3(ln ρ)N+1

ρN(81)

where c3 is a constant number which only depends on N . Thus,

limρ→∞

− logPelog ρ

≥ limρ→∞

N log ρ− log(ln ρ)N+1 − log c3

log ρ= N. (82)

According to Theorem 2, this limit can not be greater than N . Therefore,

limρ→∞

− logPelog ρ

= N. (83)

Case 2, M < N :

For the N×M matrix HH, we use the first inequality in lemma 3 to bound the probability

that dHH is too small:

Pr

{

d2 ≤ γ

ρ

}

≤ Pr

{

d2HH ≤ γc1α

2M

ρ

}

(84)

≤ βN,M

(

γc1α2M

ρ

)N

(85)

≤ βN,M

(

γc1α2M

ρ

)N

for γ > 1 and ρ >1

c1α2M

(86)

≤ c2γN

ρNfor γ > 1 and ρ >

1

c1α2M

(87)

23

.

The rest of proof is similar to the case 1.

Corollary 1: Perturbation technique achieves the maximum precoding diversity in fixed-

rate MIMO broadcast systems.

Proof: In the perturbation technique, for the transmission of each data vector u, among

the set{

H+(u + al)‖l ∈ Z2M}

, the nearest point to the origin is chosen. The transmitted

vector in the lattice-reduction-based method belongs to that set. Therefore, the energy of the

transmitted signal in the lattice-reduction-based method can not be less than the transmitted

energy in the perturbation technique. Thus, the average transmitted energy for the perturbation

method is at most equal to the average transmitted energy of the lattice-reduction-based

method. The rest of the proof is the same as the proof of theorem 3.

B. Fixed sum-rate

When the sum-rate Rsum is given, similar to the previous part, we can define the outage

probability as the probability that the sum-capacity of the broadcast system is less than Rsum.

Theorem 4: For a MIMO broadcast system with N transmit antennas, M single-antenna

receivers, and a given sum-rate Rsum,

limρ→∞

− logPoutlog ρ

≤ NM. (88)

Proof:For any channel matrix H, we have [9]

Csum = supD

log |IM + ρHHDH| (89)

where D is a diagonal matrix with non-negative elements and unit trace. Also, [29]

|2ρHHDH| ≤ (2ρtrHHDH)M

MM=

(2ρtrHHH)M

MM. (90)

The entries of H have iid complex Gaussian distribution with unit variance. Thus

trρHHH is equal to the square norm of an NM -dimensional complex Gaussian vector and

24

has a chi square distribution with 2NM degrees of freedom. Thus, we have (similar to the

equations 22-28, in the proof of theorem 2),

Pr

{

log(2ρtrHHH)

M

MM≤ Rsum

}

(91)

= Pr

{

trHHH ≤ 2RsumM M

2ρ

}

(92)

≥ 2NRsumMNMc

ρNM(for ρ >

2RsumM M

2) (93)

where c is a constant number. Now,

limρ→∞

− logPoutlog ρ

= (94)

limρ→∞

− log Pr {supD log |IM + ρHHDH| ≤ Rsum}log ρ

(95)

≤ limρ→∞

− log Pr{supD log |2ρHHDH| ≤ Rsum}log ρ

. (96)

By using (90), (93), and (96):

limρ→∞

− logPoutlog ρ

≤ limρ→∞

− log 2NRsumMNM cρNM

log ρ= NM. (97)

The slope NM for the SER curve can be easily achieved by sending to only the best user.

Similar to the proof of theorem 3, the slope of the symbol-error rate curve is asymptoticly

determined by the slope of the probability that |hmax| is smaller than a constant number, where

hmax is the entry of H with maximum norm. Due to the iid complex Gaussian distribution of

the entries of H, this probability decays with the same rate as ρ−NM , for large ρ. However,

although sending to only the best user achieves the optimum slope for the SER curve, it is

not an efficient transmission technique because it reduces the capacity to the order of log ρ

(instead of M log ρ).

25

VII. SIMULATION RESULTS

Figure 3 presents the simulation results for the performance of the proposed schemes,

the perturbation scheme [12], and the naive channel inversion approach. The number of the

transmit antennas is N = 4 and there are M = 4 single-antenna users in the system. The

overall transmission rate is 8 bits per channel use, where 2 bits are assigned to each user, i.e.

a QPSK constellation is assigned to each user (i.e. the rate is 2 bits per complex dimension

or 1 bit per real dimension).

By considering the slope of the curves in figure 3, we see that by using the proposed

reduction-based schemes, we can achieve the maximum precoding diversity, with a low

complexity. Also, as compared to the perturbation scheme, we have a negligible loss in the

performance (about 0.2 dB). Moreover, compared to the approximated perturbation method

[14], we have about 1.5 dB improvement by sending the bits, corresponding to the shift vector,

at the beginning of the transmission. Without sending the shift vector, the performance of

the proposed method is the same as that of the approximated perturbation method [14]. The

modified perturbation method (with sending two shift bits for each user) has around 0.3 dB

improvement compared to the perturbation method.

Figure 4 compares the regularized proposed scheme with V-BLAST modifications of

Zero-Forcing and Babai approximation for the same setting. As shown in the simulation

results (and also in the simulation results in [12] and [14]), the modulo-MMSE-VBLAST

scheme does not achieve a precoding diversity better than zero forcing (though it has a

good performance in the low SNR region). However, combining the lattice-reduction-aided

(LRA) scheme with MMSE-VBLAST precoding or other schemes such as regularization

improves its performance by a finite coding gain (without changing the slope of the curve

of symbol-error-rate). Combining both the regularization and the shift vector can result in

better performance compared to other alternatives.

Figure 5 compares the performances of the fixed-rate and the variable-rate transmission

using lattice-basis reduction for N = 2 transmit antennas and M = 2 users. In both cases,

the sum-rate is 8 bits per channel use (in the case of fixed individual rates, a 16QAM

constellation is assigned to each user). We see that by eliminating the equal-rate constraint,

26

8 10 12 14 16 18 20 22 2410−5

10−4

10−3

10−2

10−1

100

Sym

bol E

rror R

ate

SNR (dB)

Channel inversionPerturbation with Babai approximationProposed method without shift vectorProposed method with shift vectorPerturbation techniqueModified perturbation technique

Fig. 3. Symbol Error Rate of the proposed schemes, the perturbation scheme [12], and the naive channel inversion approach

for N = 4 transmit antennas and M = 4 single-antenna receivers with the rate R = 2 bits per channel use per user.

we can considerably improve the performance (especially, for high rates). In fact, the diversity

gains for the equal-rate and the unequal-rate methods are, respectively, M and NM . It should

be noted that the average transmit power and also the sum-rate are fixed for different channel

realizations and no long-run power allocation is used. The gain of the variable-rate originates

from allocating higher rates to users who have better channels.

VIII. CONCLUSION

A new viewpoint for designing and analysing lattice-reduction-aided communications

over MIMO broadcast channels is introduced. Lattice basis reduction helps us to reduce the

average transmitted energy by modifying the region which includes the constellation points.

It is shown by a mathematical proof that the lattice-reduction-aided precoding (which has

27

8 10 12 14 16 18 20 22 2410−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Sym

bol E

rror R

ate

4 transmit antennas, 4 single−antenna users, rate=4X2 bits per channel use

Perturbation techniqueProposed method with regularization and shift vectorBabai approximation−VBLASTZero forcingTHP−VBLAST

Fig. 4. Comparison of the regularized proposed scheme with V-BLAST modifications of Zero-Forcing and Babai

approximation (for N = 4 transmit antennas and M = 4 single-antenna receivers with the rate R = 2 bits per channel use

per user.).

a polynomial-time complexity) achieves the maximum precoding diversity. Also, simulation

results show that the performance of the proposed modified scheme is very close to the

performance of the perturbation method. Also,

APPENDIX A

In this Appendix, we compute the second moment of a parallelotope whose centroid is

the origin and its edges are equal to the basis vectors of the lattice.

Assume that A is an M -dimensional parallelotope and X is its second moment. The

second moment of 12A is

(

12

)M+2X . The parallelotope A can be considered as the union

of 2M smaller parallelotopes which are constructed by ± 12b1,±1

2b2, ....,±1

2bM , where bi,

1 ≤ i ≤ M , is a basis vector. These parallelotopes are translated versions of 12A with the

28

15 20 25 3010−5

10−4

10−3

10−2

10−1

100

Sum−Rate = 8 bits per channel use, 2 transmit antennas, 2 users

Symbol Error Rate

SNR (dB)

Fixed−rateVariable−rate

Fig. 5. Performance comparison between the fixed-rate and the variable-rate transmission for N = 2 transmit antennas

and M = 2 single-antenna receivers with sum-rate 8 bits per channel use.

translation vectors Ti = ±12b1 ± 1

2b2 ± .... ± 1

2bM , 1 ≤ i ≤ 2M . The second moments of

these parallelotopes are equal to(

12

)M+2X+‖Ti‖2Vol(1

2A), 1 ≤ i ≤ 2M . By the summation

over all these second moments, we can find the second moment of A.

X =2M∑

i=1

[

(

1

2

)M+2

X + ‖Ti‖2.Vol(1

2A)

]

(98)

=

(

1

2

)2

X + 2M−2(‖b1‖2 + ...+ ‖bM‖2).Vol(1

2A) (99)

=1

4X +

1

4(‖b1‖2 + ...+ ‖bM‖2).Vol(A) (100)

=⇒ X =1

3(‖b1‖2 + ... + ‖bM‖2).Vol(A). (101)

29

APPENDIX B

PROOF OF LEMMA 3

Lemma 3 states that the probability that a lattice, generated by M independent N -

dimensional complex Gaussian vectors, N ≥ M , with a unit variance per each dimension,

has a nonzero point inside a sphere (centered at origin and with the radius ε) is bounded by

βN,Mε2N for N > M , and βN,Mε2N max

{

(− ln ε)N+1, 1}

for N = M ≥ 2. We can assume

that ε < 1 (for ε ≥ 1, lemma 3 is trivial because the probability is bounded).

A. Case 1: M = 1

When M = 1, the lattice consists of the integer multiples of the basis vector v. If the

norm of one of these vectors is less than ε, then the norm of v is less than ε. Consider

the variance of the components of v as %2. The vector v has an N -dimensional complex

Gaussian distribution, fv(v). Therefore, the probability of this event is,

Pr {‖v‖ ≤ ε} =

∫

‖v‖≤εfv(v) dv ≤

∫

‖v‖≤ε

1

πN%2Ndv ≤ βN,1

ε2N

%2N. (102)

When the variance of the components of v is equal to one, we have,

Pr {‖v‖ ≤ ε} ≤ βN,1ε2N . (103)

B. Case 2: N > M > 1

Consider L(v1 ,...,vM) as the lattice generated by v1,v2,...,vM . Each point of L(v1,...,vM)

can be represented by v(z1,...,zM) = z1v1 + z2v2 + ...+ zMvM , where z1, ..., zM are complex

integer numbers. The vectors v1,v2,...,vM are independent and jointly Gaussian. Therefore,

for every integer vector z = (z1, ..., zM), the entries of the vector v(z1,...,zM) have complex

Gaussian distributions with the variance

%2z = ‖z‖2%2 =

(

|z1|2 + ...+ |zM |2)

%2. (104)

Therefore, according to the lemma for M = 1,

Pr{

‖v(z1,...,zM)‖ ≤ ε}

≤ βN,1ε2N

(|z1|2 + ...+ |zM |2)N. (105)

30

Now, by using the union bound,

Pr {dH ≤ ε} ≤∑

z6=0

Pr{

‖v(z1,...,zM)‖ ≤ ε}

(106)

≤∑

z6=0

βN,1ε2N

(|z1|2 + ... + |zM |2)N(107)

= βN,1

∑

1≤‖z‖<2

ε2N

‖z‖2N+

∑

2≤‖z‖<3

ε2N

‖z‖2N

+∑

3≤‖z‖<4

ε2N

‖z‖2N+ ...

. (108)

The M -dimensional complex integer points z = (z1, ..., zM), such that k ≤ ‖z‖ < k+1,

can be considered as the centers of disjoint unit-volume cubes. All these cubes are inside the

region between the 2M -dimensional spheres, with radii k−1 and k+2. Therefore, the number

of M -dimensional complex integer points z = (z1, ..., zM), such that k ≤ ‖z‖ < k + 1, can

be bounded by the volume of the region between these two 2M -dimensional spheres. Thus,

this number is bounded by c1k2M−1 for some constant5 c1. Therefore,∑

k≤‖z‖<k+1

ε2N

‖z‖2N≤ c1k

2M−1 ε2N

k2N(109)

(108), (109) =⇒ Pr {dH ≤ ε} ≤ c1βN,1ε2N + 22M−1c1βN,1

ε2N

22N+

+32M−1c1βN,1ε2N

32N+ ... (110)

≤ c1βN,1ε2N

∞∑

k=1

1

k2N−2M+1. (111)

According to the assumption of this case, N > M ; hence, 2N−2M+1 ≥ 2. Therefore,

the above summation is convergent:

Pr {‖v‖ ≤ ε} ≤ βN,Mε2N . (112)

5Throughout this proof, c1, c2, ... are some constant numbers.

31

...................

ε/2

z = 2z = 3

z = 1

z = 4z = −n

0

radius = ε/n

v

Sv−n Sv,n

Sv,4

Sv,3

Sv,2

S0

z = n

center = v/n

radius = εcenter = 0

Sv,1 = Sv

Fig. 6. The family of spheres Sv,z

C. Case 3: N = M > 1

Each point of L(v1,...,vN) can be represented by zvN −v, where v belongs to the lattice

L(v1,...,vN−1) and z is a complex integer. Consider Sv as the sphere with radius ε and centered

at v. Now, zvN − v belongs to S0 iff the zvN belongs to Sv. Also, the sphere Sv includes

a point zvN iff Sv,z includes v , where Sv,z = Svz

is the sphere centered at v/z with radiusε

|z| (see figure 6). Therefore, the probability that a lattice point exists in Sv is equal to the

probability that vN is in at least one of the spheres {Sv,z}, z 6= 0.

If we consider dH as the minimum distance of L(v1,...,vN) and R as an arbitrary number

greater than 1:

32

Pr {dH ≤ ε} = Pr{(

L(v1 ,...,vN) − 0)

∩ S0 6= ∅}

= Pr

{

vN ∈⋃

v

⋃

z 6=0

Sv,z

}

(113)

≤ Pr

vN ∈⋃

‖vz‖≤RSv,z

+ Pr

vN ∈⋃

‖vz‖>RSv,z

(114)

In the second term of (114), all the spheres have centers with norms greater than R and radii

less than 1 (because |z| ≥ 1). Therefore,⋃

‖vz‖>RSv,z ⊂ {x | ‖x‖ > R− 1} (115)

(115) =⇒ (114) ≤ Pr

vN ∈⋃

‖vz‖≤RSv,z

+ Pr {‖vN‖ > R− 1} (116)

≤ Pr

vN ∈⋃

‖vz‖≤R,|z|≤ε1−N

Sv,z

+ Pr

vN ∈⋃

‖vz‖≤R,|z|>ε1−N

Sv,z

+ Pr {‖vN‖ > R− 1} .

(117)

We bound the first term of (117) as the following:

Pr{vN ∈⋃

‖vz‖≤R,|z|≤ε1−N

Sv,z} (118)

≤

∑

‖v‖≤2R

∑

|z|≥1

Pr{vN ∈ Sv,z}+∑

2R<‖v‖≤3R

∑

|z|≥2

Pr{vN ∈ Sv,z}+

... +∑

bε1−N cR<‖v‖≤(bε1−N c+1)R

∑

|z|≥bε1−NcPr{vN ∈ Sv,z}

. (119)

Noting that the pdf of vN is less than or equal to 1πN

,

(118) ≤ 1

πN

∑

‖v‖≤2R

∑

|z|≥1

Vol(Sv,z) +∑

2R<‖v‖≤3R

∑

|z|≥2

Vol(Sv,z)+

...+∑

bε1−N cR<‖v‖≤(bε1−N c+1)R

∑

|z|≥bε1−NcVol(Sv,z)

(120)

33

≤ 1

πN

∑

‖v‖≤2R

∑

|z|≥1

c2ε2N

|z|2N +∑

2R<‖v‖≤3R

∑

|z|≥2

c2ε2N

|z|2N +

...+∑

bε1−N cR<‖v‖≤(bε1−N c+1)R

∑

|z|≥bε1−Nc

c2ε2N

|z|2N

. (121)

By using (109), for one-dimensional complex vector z = z,∑

|z|≥i

c2ε2N

|z|2N =

∞∑

k=i

c2.∑

k≤|z|≤k+1

ε2N

|z|2N ≤∞∑

k=i

c1c2ε2N

k2N−1≤ c3ε

2N

i2N−2(122)

Now,

(122) =⇒ (118) ≤ 1

πN

∑

‖v‖≤2R

c3ε2N +

∑

2R<‖v‖≤3R

c3ε2N

22N−2+

...+∑

bε1−N cR<‖v‖≤(bε1−N c+1)R

c3ε2N

bε1−Nc2N−2

. (123)

Assume that the minimum distance of L(v1 ,...,vN−1) is dN−1. The spheres with the radius

dN−1/2 and centered by the points of L(v1,...,vN−1) are disjoint. Therefore, the number of

points from the (N − 1)-dimensional complex lattice L(v1,...,vN−1), such that ‖v‖ ≤ 2R, is

bounded by c4(2R+dN−1/2)2N−2

d2N−2N−1

(it is bounded by the ratio between the volumes of (2N − 2)-

dimensional spheres with radii 2R+ dN−1/2 and dN−1/2). Also, the number of points from

L(v1,...,vN−1), such that (k−1)R < ‖v‖ ≤ kR, is bounded by c4(kR)2N−3(R+dN−1)

d2N−2N−1

(it is bounded

by the ratio between the volumes of the region defined by (k − 1)R − dN−1/2 < ‖x‖ ≤kR + dN−1/2 and the sphere with radius dN−1/2):

(123) ≤ c5(2R + dN−1/2)2N−2

d2N−2N−1

.ε2N +c5R

2N−3(R + dN−1)

d2N−2N−1

.ε2N

bε1−N c∑

k=2

1

k(124)

(123) ≤ c5(2R + dN−1/2)2N−2

d2N−2N−1

.ε2N +c5R

2N−3(R + dN−1)

d2N−2N−1

.ε2N . ln(ε1−N) (125)

≤ c6ε2N .max

(

R2N−2

d2N−2N−1

, 1

)

.max {− ln ε, 1} . (126)

34

According to the proof of the case 2, we have Pr {dN−1 ≤ η} ≤ βN,N−1η2N . Therefore,

EdN−1

{

max

(

R2N−2

d2N−2N−1

, 1

)}

(127)

≤ 1.Pr {dN−1 > R}+22N−2.Pr

{

1

2R < dN−1 ≤ R

}

+32N−2.Pr

{

1

3R < dN−1 ≤

1

2R

}

+...

(128)

≤ 1 + 22N−2.Pr {dN−1 ≤ R}+ 32N−2.Pr

{

dN−1 ≤1

2R

}

+ ... (129)

≤ 1 +∞∑

k=1

(k + 1)2N−2

k2N.R2NβN,N−1 ≤ c7R

2N (130)

=⇒ EdN−1

{

c6ε2N .max

(

R2N−2

d2N−2N−1

, 1

)

.max {− ln ε, 1}}

≤ c8ε2N .R2N .max {− ln ε, 1} .

(131)

To bound the second term of (117), we note that for |z| ≥ ε1−N , the radii of the spheres

Sv,z are less or equal to εN , and the centers of these spheres lie on the (N − 1)-dimensional

complex subspace containing Lv1,...,vN−1. Also, the norm of these centers are less than R.

Therefore, all of these spheres are inside the region A which is an orthotope centered at the

origin, with 2N real dimensions (see figure 7):

⋃

‖vz‖≤R,|z|>ε1−N

Sv,z ⊂ A (132)

=⇒ Pr{vN ∈⋃

‖vz‖≤R,|z|>ε1−N

Sv,z} ≤1

πNVol

⋃

‖vz‖≤R,|z|>ε1−N

Sv,z

≤ 1

πNVol(A) (133)

≤ 1

πN(2εN)2(2R + εN)2N−2 (134)

35

0

Subspace containing Lv1,...,vN−1

2εN

2R + 2εN

2R + 2εN

Fig. 7. The orthotope A

Also, according to the Gaussian distribution of the entries of vN (which have Variance12

on each real dimension), we can bound the third term of (117) as,

Pr{‖vN‖ > R − 1} ≤ 2NQ

(√

R− 1

N

)

≤ c9e−

“

R−1√2N

”2

. (135)

By using (131), (134), and (135),

Pr {dH ≤ ε} ≤ c8ε2N .R2N .max {− ln ε, 1}+ 1

πN(2εN)2(2R+εN)2N−2+c9e

−“

R−1√2N

”2

. (136)

The above equation is true for every R > 1. Therefore, using R =√

2N√

− ln(ε2N) + 1,

Pr {dH ≤ ε} ≤ βN,Nε2N .max

{

(− ln ε)N+1, 1}

. (137)

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